For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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12
votes
4answers
339 views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
12
votes
2answers
311 views

Geometric interpretation of the cofactor expansion theorem

I find the geometric interpretation of determinants to be really intuitive - they are the "area" created by the column vectors of the matrix. Could someone give me a geometric interpretation of the ...
12
votes
2answers
510 views

Why is the identity the only symmetric $0$-$1$ matrix with all eigenvalues positive?

While thinking about this question I managed to convince myself that the identity is the only symmetric $0$-$1$ matrix with all eigenvalues positive. However, the argument is rather low-level. It ...
12
votes
5answers
3k views

Relation between Cholesky and SVD

When we have a symmetric matrix $A = LL^*$, we can obtain L using Cholesky decomposition of $A$ ($L^*$ is $L$ transposed). Can anyone tell me how we can get this same $L$ using SVD or Eigen ...
12
votes
3answers
488 views

Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible

I came across the following problem that says: Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible. Then how can I prove the following: rank $A$+ ...
12
votes
1answer
760 views

Effect of elementary row operations on determinant?

1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ...
12
votes
1answer
3k views

Is it faster to multiply a matrix by its transpose than ordinary matrix multiplication?

I'm writing a program that multiples a matrix by its transpose, and was trying to find efficiency hacks I could exploit considering that the two matrices being multiplied are related. Any ideas?
12
votes
2answers
602 views

Properties of 4 by 4 Matrices

Define $ A=\begin{pmatrix} x_1 & x_2 & 0 & 0\\ 0 &1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}, B=\begin{pmatrix} 1 & 0 & 0 & 0\\ x_3 ...
12
votes
1answer
150 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
12
votes
2answers
354 views

Product of nilpotent matrices.

Let $A$ and $B$ be $n \times n$ complex matrices and let $[A,B] = AB - BA$. Question: If $A , B$ and $[A,B]$ are all nilpotent matrices, is it necessarily true that $\operatorname{trace}(AB) ...
12
votes
1answer
104 views

Check membership in a matrix group

I'm looking for a (preferably somewhat efficient) algorithm for this problem: Given a normal subgroup of $SL(m, \mathbb{Z})$ generated by a finite set $\{M_1, M_2, \dotsc, M_n\}$, and some $A \in ...
12
votes
2answers
503 views

The infinite-dimensional limit of sequence of solutions of linear equations when the number of equations goes to infinity

Suppose we have an infinite-dimensional real vector $y=(y_1,...)$. Suppose we have an infinite-dimensional real matrix $C=(c_{ij})$, $i,j\in\mathbb{N}$. Let $C^k$ be a submatrix of $C$, ...
12
votes
2answers
550 views

A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from “Topics in Matrix Analysis” for matrices of algebraic numbers.

It is a well known fact that if $A,B\in M_{n\times n}(\mathbb C)$ and $AB=BA$, then $e^Ae^B=e^Be^A.$ The converse does not hold. Horn and Johnson give the following example in their Topics in Matrix ...
12
votes
1answer
371 views

Eigenvalues of $A$ and $A + A^T$

This question has popped up at me several times in my research in differential equations and other areas: Let $A$ be a real $N \times N$ matrix. How are the eigenvalues of $A$ and $A + A^T$ related? ...
12
votes
1answer
324 views

A sharper bound for $\|\cos(kA)\|_{\infty}$ for symmetric stochastic matrices

Given $A \in \mathbb{R}^{n \times n}$ that is symmetric, stochastic and diagonalizable, and $k \in \mathbb{N}$, I am interested in bounding $\|\cos(kA)\|_{\infty}$ from above. $\| \|_{\infty}$ is ...
12
votes
3answers
465 views

Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
11
votes
6answers
1k views

Matrix to power $2012$

How to calculate $A^{2012}$? $A = \left[\begin{array}{ccc}3&-1&-2\\2&0&-2\\2&-1&-1\end{array}\right]$ How can one calculate this? It must be tricky or something, cause there ...
11
votes
6answers
752 views

A matrix satisfying $AB-BA=B$

If $A$ and $B$ are two matrices of $\mathcal{M}_n(\mathbb{R}$) such that $$AB-BA=B$$ how can we prove that $B$ isn't invertible? my attempt: I found that $\mathrm{tr}(B)=0$ but I know that this is ...
11
votes
4answers
349 views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
11
votes
6answers
348 views

Given a matrix $A$, find $A^n$

Given the matrix $$A = \left[{9\atop20}{-4\atop-9}\right]$$ how do I find $A^7$ or $A^{54}$ or $A^{2008}$ (etc.) ? I know I need the eigenvalues of A, but I'm not sure what to do afterwards. Is the ...
11
votes
4answers
568 views

What makes elementary row operations “special”?

This is probably a stupid question, but what makes the three magical elementary row operations, as taught in elementary linear algebra courses, special? In other words, in what way are they "natural" ...
11
votes
1answer
4k views

Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!)

First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an ...
11
votes
2answers
467 views

Characteristic polynomials exhaust all monic polynomials?

Let $A$ be an $n\times n$ matrix, then $\mathrm{char}_A(x):=\det(xI-A)$ is a monic polynomial of degree $n$. It is called the characteristic polynomial of $A$. My question is the converse: Let ...
11
votes
6answers
289 views

Maximizing the sum $\sum\limits_{i=1}^nx_ix_{i+1}$ subject to $\sum\limits_{i=1}^nx_i=0$ and $\sum\limits_{i=1}^nx_i^2=1$

Is there an efficient way of solving the following problem? Given $x_i\in \mathbb R$, and that $\sum\limits_{i=1}^nx_i=0$ and $\sum\limits_{i=1}^nx_i^2=1$. I want to maximize ...
11
votes
3answers
2k views

Recovering the two SU(2) matrices from SO(4) matrix

Since there is a 2-1 homomorphism from SU(2)xSU(2) to SO(4) there should be a way to recover the two SU(2) matrices given an SO(4) matrix. I believe I could set this up as a system of equations ...
11
votes
4answers
252 views

Finding all matrices $B$ such that $AB=BA$ for a fixed matrix $A$

Let $$ A=\begin{pmatrix} 1 & 0 & 0 \\ 0& 1 & 0 \\ 3 & 1 & 2 \end{pmatrix} $$ Find all matrices $B$ such that $AB=BA$. Attempt at solution: I can show that $A$ is ...
11
votes
2answers
691 views

Why are the eigenvalues of these “bitwise XOR matrices” integers?

In the course of playing around with this question, I have hit upon a question of my own. Consider the $n\times n$ symmetric matrix $\mathbf X$ whose entries are given by ...
11
votes
3answers
404 views

Eigenvalues of some peculiar matrices

While I was toying around with matrices, I chanced upon a family of tridiagonal matrices $M_n$ that take the following form: the superdiagonal entries are all $1$'s, the diagonal entries take the form ...
11
votes
5answers
310 views

Why are there$ 736$ $2\times 2$ matrices $(M)$ over $\mathbb{Z}_{26}$ for which it holds that $M=M^{-1}$?

I'm currently trying to introduce myself to cryptography. I'm reading about the Hill Cipher currently in the book Applied Abstract Algebra. The Hill Cipher uses an invertible matrix $M$ for ...
11
votes
2answers
803 views

Block Diagonal Matrix Diagonalizable

I am trying to prove that: The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diaginalizable, if only if $A$ and $B$ are diagonalizable. If $A\in\mathbb{C}^n$ ...
11
votes
1answer
872 views

4 by 4 Matrix Puzzle

I was solving the puzzle for the Company interview exam. I found this puzzle, I cannot come up with the solution. How to solve it and what is the correct answer? Determine the number of $4\times ...
11
votes
1answer
1k views

Probability that a random binary matrix is invertible?

What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible? Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$. Is there an explicit formula as a ...
11
votes
2answers
3k views

Determine the matrix relative to a given basis

Question: (a) Let $f: V \rightarrow W$ with $ V,W \simeq \mathbb{R}^{3}$ given by: $$f(x_1, x_2, x_3) = (x_1 - x_3, 2x_1 -5x_2 -x_3, x_2 + x_3).$$ Determine the matrix of $f$ relative to the basis ...
11
votes
1answer
119 views

Prove a $n \times n $ matrix has rank 3

I have been examining a problem dealing with finding the rank of a $n \times n $ matrix $M$ as follows: \begin{bmatrix} 0&1&4&9&16&\cdots &(n-1)^2\\ ...
11
votes
1answer
955 views

The Determinant of a Sum of Matrices

Given $N$ $n \times n$ matrices $\mathsf{A}^{1}, \dots, \mathsf{A}^{N}$, \begin{align} \det \left( \sum_{i = 1}^{N} \mathsf{A}^{i} \right) = \sum_{\sigma \in S} \det \mathsf{A}^{\sigma}, \end{align} ...
11
votes
6answers
17k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
11
votes
1answer
72 views

Largest determinant of a real $3\times 3$-matrix

What is the largest determinant of a real $3\times 3$-matrix with entries from the interval $[-1,1]$ ? A result of John Williamson says that the largest value is equal to $4$, if the entries are just ...
11
votes
1answer
464 views

Direct proof that nilpotent matrix has zero trace

Does anyone know a proof from the first principles that a nilpotent matrix has zero trace. No eigenvalues, no characteristic polynomials, just definition and basic facts about bases and matrices.
11
votes
4answers
3k views

Square root of a matrix

Under what conditions does a matrix $A$ have a square root? I saw somewhere that this is true for Hermitian positive definite matrices(whose definition I just looked up). Moreover, is it possible ...
11
votes
3answers
222 views

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
11
votes
2answers
254 views

Determinant of a finite-dimensional matrix in terms of trace

I have noticed that for the case of 1x1, 2x2 and 3x3 matrices $A$, $B$, I can write the determinant of their commutator $C=[A,B]$ in terms of traces: 1x1 matrices $A$, $B$: $$\det(C)=\text{tr}(C)$$ ...
11
votes
3answers
247 views

Prove the inequality: $\prod_{j=1}^ka_{jj}\leq\left(\frac{1}{k}\sum_{j=1}^k\lambda_j\right)^k.$

To all those who are eagerly awaiting a new question, all those who love math, I give this challenge and I hope for you good moments of reflection. Let $A=(a_{ij})_n$ a real nonnegative symmetric ...
11
votes
1answer
212 views

Proving a certain determinant $\left|\det A\right|$ is complete square

Consider the following matrix $$ A_{ij}= \begin{cases} 1\quad\text{ if }\space (i+j)\space\text{ is prime,}\\ 0\quad\text{ otherwise.} \end{cases} $$ How can one prove that $\left|\det A\right|$ is a ...
11
votes
1answer
173 views

how prove the following statment for this matrix.

Let $A:=[a_{ij}]_{n×n}$ , $a_{ij}=0$ or $a_{ij}=1$ and $\exists m \in\mathbb N$ such that $A^m=J-I$, where $I$ is the identity matrix and $J=[1]_{n×n}$ (each entry is $1$). How to prove: $\exists ...
11
votes
2answers
423 views

Does this expression have a (“better”) determinant form?

[Edit I've found a determinant that satisfies the letter of the previous version of this question, but not its "Cayley-Menger" spirit.] A tetrahedron with face areas $w$, $x$, $y$, $z$ and ...
11
votes
1answer
467 views

spectral norm of random matrix

Suppose $A$ is a $n \times n$ random matrix with centered Gaussian (real) i.i.d entries with variance $\sigma^2/n$. What to we know about the spectral norm $s(A)$ of $A$, that is $\sqrt{\rho(A^t A)}$ ...
11
votes
1answer
604 views

Determinant of a symmetric matrix

Given an $n\times n$ matrix $C= [c_{ij}]$ which is symmetric (i.e. $c_{ij}=c_{ji}\ \forall i,j$) calculate the determinant of the following matrix (assume $c_{ij} \neq 0\ \forall i,j$): ...
11
votes
1answer
206 views

Characterization of Volumes of Lattice Cubes

Here is a problem that came up in a conversation with a professor. I do not know if he knew the answer (and told me none of it) and has since passed so I can no longer ask him about it. Let $C$ be a ...
11
votes
0answers
253 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $V= \begin{bmatrix} 1 ...
11
votes
0answers
363 views

Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 ...